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ISI B.Math Entrance Exam
2010 ISI B.Math Entrance Exam
9
9
Part of
2010 ISI B.Math Entrance Exam
Problems
(1)
I.S.I. B.Math.(Hons.) Admission test : 2010 Problem 9
Source:
4/11/2012
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial with integer co-efficients. Assume that
3
3
3
divides the value
f
(
n
)
f(n)
f
(
n
)
for each integer
n
n
n
. Prove that when
f
(
x
)
f(x)
f
(
x
)
is divided by
x
3
ā
x
x^3-x
x
3
ā
x
, the remainder is of the form
3
r
(
x
)
3r(x)
3
r
(
x
)
where
r
(
x
)
r(x)
r
(
x
)
is a polynomial with integer coefficients.
algebra
polynomial